Title: Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry Show Chinese title

Title: 周期晶体中的对称性和局域化：具有费米子时间反演对称性的布洛赫束的平凡性

Abstract: We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The role of additional $\mathbb{Z}_2$-symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same $\mathbb{Z}_2$-symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators. Show Chinese abstract

Abstract: 我们描述了群论和丛论方法在固态物理学中的若干应用，展示出对称性如何导致间隙晶体固体（如绝缘体和半导体）中电子局域化的证明。 我们简要回顾了周期算子的布洛赫-弗洛凯分解及其相关的布洛赫框架和复合威尔纳函数的概念。 我们表明，后者的几乎指数局域化当且仅当存在一个光滑周期布洛赫框架，并且后者条件的障碍是一个厄米向量丛（称为布洛赫丛）的平凡性。 讨论了额外的$\mathbb{Z}_2$对称性（如时间反演对称性和空间反射对称性）的作用，展示了时间反演对称性如何导致布洛赫丛在玻色子和费米子情况下的平凡性。 此外，同样的$\mathbb{Z}_2$对称性允许定义一种更精细的同构概念，并由此定义新的拓扑不变量，这些不变量与福（Fu）、卡内（Kane）和梅勒（Mele）在拓扑绝缘体背景中引入的指标一致。